accurate volume calculation driven by delaunay

Research ArticleAccurate Volume Calculation Driven by DelaunayTriangulation for Coal Measurement

Yong Liu 1 and Yanwei Zheng 2

1Huadian Zibo ermal Power Co Ltd Zibo 255054 China2School of Computer Science and Technology Shandong University Qingdao 266237 China

Correspondence should be addressed to Yanwei Zheng zhengywsdueducn

Received 29 December 2020 Revised 12 March 2021 Accepted 8 April 2021 Published 16 April 2021

Academic Editor Antonio J Pentildea

Copyright copy 2021Yong Liu and Yanwei Zhengis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Volume calculation from 3D point cloud is widely used in engineering and applications e existing methods either have largeerrors or are time-consuming is paper focuses on the coal measurement Based on the triangular mesh generated from thepoint cloud each triangle is projected downward to the base plane to form a voxel We derive the calculation formula of voxel byan integral method which is more efficient than the method of decomposing voxel into tetrahedrons and more accurate thanslicing methods Furthermore this paper proposes a Delaunay triangulation-driven volume calculation (DTVC) method DTVCdoes not preserve the Delaunay triangles but directly calculates the volume in the process of triangulation It saves memory andrunning time Experimental results show that DTVC has achieved a good balance between error and efficiency

1 Introduction

Volume calculation is to compute the space occupied by anobject through point cloud that generated from 3D re-construction Volume calculation underpins many crucialapplications such as the volume measurement of coal [1]ore [2] earthwork [3] and tree [4 5] and the disease di-agnosis in medical field [6 7] is paper aims to the coalmeasurement where volume calculation has been a chal-lenging task due to the difficulty in determining the objectboundary from the scatters of point cloud

A rough method to calculate object volume is to fit astandard geometry such as ellipsoid [8] and cylinder [9ndash11]ese methods are fast and simple because they only need tofit the geometry parameters However due to the differencebetween the actual object and the regular geometry thecalculated volume error is large

e more accurate calculation methods can be roughlydivided into three categories (1) Convex hull method[12 13] Irregular object is represented by convex hull econvex hull can be decomposed into two triangular meshese corresponding projection volume is calculated by the

orthographic projection method and the difference is theobject volume e method is suitable for the convex modeland the error of nonconvex model is large (2) Slicingmethod [14 15] is method slices the point cloud using aplane that is perpendicular to a coordinate axis e totalvolume was obtained by accumulating the volume of eachslice e thickness of the slice has a great influence on theresult (3) Projection method [16] is method has twostages First the 2D projection of 3D point cloud is meshedusually using the triangulation grid Each triangle in themesh projects to a plane e projected triangle and theoriginal triangle form a pentahedron Second the penta-hedron volume is calculated and the total volume is ob-tained by accumulating the volume of all the pentahedrons

Coal pile is not a convex model as shown in Figure 1 sothe convex hull method cannot be used in this applicatione top slice of coal pile has many independent areas It isdifficult to split them so the error is large when the slicingmethod is usedus the triangulation projection method issuitable for coal measurement

ere are two methods in the pentahedron volumecalculation in triangulation projection method (1) Connect

HindawiScientific ProgrammingVolume 2021 Article ID 6613264 10 pageshttpsdoiorg10115520216613264

the diagonal lines of each pentahedron face and then thepentahedron is divided into three tetrahedrons e volumeof pentahedron is obtained by accumulating the volumes oftetrahedrons [17] (2) e pentahedron is approximated to atriangular prism to calculate the volume [18]ese methodshave large error or are time-consuming

is paper proposes a Delaunay triangulation-drivenvolume calculation (DTVC)method where the two stages inthe triangulation projection method are merged togetherWith the construction of triangular mesh the volume cal-culation is completed e pentahedron volume calculationis also improved It is more accurate and efficient than othermethods

We summarize our contributions as follows

(i) An accurate and direct method is proposed tocalculate the volume of the pentahedron whichprovides the analytic solutions of pentahedron us-ing the integral method

(ii) We also combined the two stages to one for thetriangulation projection method e volume cal-culation is driven by the triangulation

(iii) e calculation error of the whole point cloudvolume is less than the slicing method and theefficiency is higher than the existing projectionmethod e proposed method has achieved a goodbalance between error and efficiency

e remainder of this paper is organized as followsSection 2 discusses the related work about the volumecalculation of point cloud In Section 3 the accurate anddirect pentahedron volume calculation is presented InSection 4 the triangulation-driven volume calculation isintroduced Section 5 provides the experimental resultsSection 6 concludes this paper

2 Related Work

is section discusses the volume calculation methods

21 RoughMethods Guo et al [4] calculate the standing treevolume using an empirical formula where the volume is theproduct of the cross-sectional area at breast height theheight plus three and a regulatory factor Wu et al [19] andOkinda et al [8] regard the jujubes and eggs as ellipsoids tocalculate the volume In [11] eachmass of food on the dish islooked as a cylinder In [9 10] the tank is approximated to a

cylinder Hadhazi et al [7] place 16 simulated small ellipticalnodules into an anthropomorphic chest phantom to cal-culate the lung nodule volume Because the real objects arenot standard geometry the error is very large for thesemethods

22 Convex Hull Method In 1972 Graham proposed theconvex hull method [12] Speakman and Averkov [20] in-troduce the convex hull volume formula of a trilinearmonomial Wu et al [21] discuss the properties and com-putation of the boundary of convex hulls of cosphericalcircles Alshamrani et al [22] filter all points inside a four-vertex polygon to reduce the computational cost for com-puting a convex hull for a large set of points Zhao et al [23]introduce an incremental convex hull calculation and a fastreplacement for nondominated sorting Auat Cheein andGuivant [24] split the 3D raw data into subsets which havethe same convex hull Calculating the convex hull fromsubsets can decrease the computational cost of the entireprocess Kim et al [25] propose a precise convex hullcomputation method for free form models where a Gaussmap organized in a hierarchy of normal pyramids and aCoons bounding volume hierarchy are prebuilt

In application to calculate the object volume on aconveyor belt Song and Ou [26] apply a convex hull al-gorithm to find the set of the smallest convex polygon thatcontains all the object point cloud Furthermore they find arectangle which has the smallest area to determine the objectsize Bandi et al [27] also use the convex hull method tocalculate the volume of dense point cloud Xu and Xu [13]use the incremental algorithm to calculate the convex hull ofthe point cloud to approximate the object Shah et al [28]apply convex hull to calculate the representative element ofvolume by simultaneously considering the two main mac-roscopic petrophysical parameters porosity and single-phase permeability

23 SlicingMethod Chang et al [29] cut the point cloud intoslices of equal thickness along the z-axis en divide eachslice along the y-axis equally and cut each slice into sub-intervals along the x-axis Using the minimum and maxi-mum coordinates in each subinterval to calculate the area ofeach sliceen the slice areas are integrated along the z-axisto estimate the volume of the object Li et al [30] take aframe in the structured light vision as a slice Heckel et al[31] propose a segmentation-based partial volume analysis

Figure 1 Coal pile from different view positions It is not a convex model

2 Scientific Programming

method to measure the volume of solid Avdal et al [32]estimate the flow volume by the maximum velocity in eachvoxel Lyra et al [33] add transaxial slices to measure thethyroid volume of hyperthyroid patients Yu et al [34]calculate the difference in the current depth image andreference depth image by splitting planes to estimate thevolume of chest wall

In [35] a slice is called a voxel which represents aparticular space that has been divided into a grid of iden-tically sized and equally spaced cubes Basher et al [6] splitthe hippocampus into voxels A convolutional neural net-work (CNN) model is designed to predict the number ofvoxels attributed to the hippocampus and the number ofestimated hippocampal voxels is multiplied by the voxelvolume to measure the discrete volume of the hippocampusGao et al [36] use the voxel-based partial volume analysismethod to calculate the volume To simulate the voxelvolumes of fibres Aronsson [37] defines each fibre as aspline curve with an elliptical cross-sectional shape and aconstant twist per length unit

24 Projection Method Potena et al [38] adopt a tailoredmeshing strategy that performs a leaf-based clustering toobtain an approximated canopy estimation and an iterativecluster connection e volume is estimated by the resultingmesh In volume estimation of nongeometric shape cavityToumpaniaris et al [39] design the distances for the tri-angulation are from a random point of the cavity to the innerwall e rand point and the triangle form a triangularpyramid e volume summation of the triangular pyramidsis the volume of the irregular body Cai et al [40] measurethe potato volume using laser triangulation in cylindricalcoordinates

Hu et al [17] construct the mesh of 3D point cloud byusing implicit B-spline e triangles of each facet areprojected to the xOy plane and the projected triangles arecombined with the corresponding facet triangles to form apentahedron e pentahedron is divided into three tetra-hedrons by connecting the diagonals of each surface enthe pentahedron volume is computed by summarizing thetetrahedron volumes In [16] the 3D liver model which isconstructed by the marching cube algorithm is composed ofa number of triangle meshes and the normal vectors ofdifferent directions of all triangle meshes are modifiedtraversed and projected onto a projection plane of 3Dspatial coordinates one by one en a lot of pentahedronscan be built and the volumes of all pentahedrons are cal-culated by the subdivision method and the final volume of3D liver model is the algebraic sum of all pentahedronvolumes Wang et al[18] transform the pentahedron ap-proximately into a triangular prism to calculate the volumeTaking the triangle as the smallest unit the space irregularbody is projected to the xOy plane and the area of theprojected triangle is calculated by using the Helen formulae volume of the Mitsubishi column can be obtained bytaking the average value of the Z coordinate value of theprojection point as the height

3 Accurate Volume Calculation Method

is section describes the detail of accurate volume calcu-lation method which is based on the triangulation pro-jection method

31 Volume CalculationMethod e coal pile is stacked onthe horizontal ground e ground is called the base planee 3D point cloud generated from the laser scanner orcamera array is the set of discrete surface points Using thetriangulation method the messy surface points can beconnected to form a triangular mesh as shown in Figure 2

We choose a triangle ABC in the triangular mesh andproject it to the base plane to get AprimeBprimeCprime e trianglesABC and AprimeBprimeCprime and the quadrilaterals ABBprimeAprimeBCCprimeBprime and ACCprimeAprime form a pentahedron which is calledvoxel in this paper

Suppose the voxel set is σi1113864 1113865n

i1 and the volume of σi is viand then the coal volume can be computed as follows

V 1113944n

i1vi (1)

32 Equation of the Top Surface e equation of the topsurface of σi is required before calculating the volume usingintegral method Take the base plane as the xOy plane and z-axis pointing upward to establish the coordinate systemSuppose the top and bottom surfaces of voxel σi areAiBiCi

and AiprimeBiprimeCiprime respectively e coordinates of Ai Bi and Ci

are (xi1 yi1 zi1) (xi2 yi2 zi2) and (xi3 yi3 zi3)respectively

We rearrange the order of Ai Bi and Ci so thatxi1 lexi2 lexi3 From the view of the positive direction of z-axis Bi can be on the right or left side of AiCi as shown inFigure 3 We can calculate the normal vector of the topplane as shown in equation (2) and is abbreviated as inequation (3)

nirarr

irarr

jrarr

krarr

xi2 minus xi1 yi2 minus yi1 zi2 minus zi1


111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(2)

nirarr

(a b c) (3)

where a (yi2 minus yi1)(zi3 minus zi1) minus (yi3 minus yi1)(zi2 minus zi1)b (xi3 minus xi1)(zi2 minus zi1) minus (xi2 minus xi1)(zi3 minus zi1) andc (xi2 minus xi1)(yi3 minus yi1) minus (xi3 minus xi1)(yi2 minus yi1)

InAiprimeBiprimeCiprime Aiprime Biprime and Ci

prime are not collinear so Ai Bi andCi are not in a plane that is perpendicular to the base planeus cne 0

e equation of the top plane can be expressed as

a x minus xi1( 1113857 + b y minus yi1( 1113857 + c z minus zi1( 1113857 0 (4)

which can be transformed as follows

z aprimex + bprimey + cprime (5)

Scientific Programming 3

where aprime minus (ac) bprime minus (bc) and cprime (ac)xi1 + (bc)yi1+zi1

33 e Edge Equation of the Projected Triangle Lookingfrom the positive to negative direction of z-axis Bi

prime can be onthe right or left of line Ai

primeCiprime as shown in Figures 3(a) and

3(b) respectivelyey are corresponding to the projectionsas shown in Figures 4(a) and 4(b) respectively

e coordinates of Aiprime Biprime and Ci

prime are (xi1 yi1 0)(xi2 yi2 0) and (xi3 yi3 0) respectively We first onlyconsider the case of xi1 lt xi2 lt xi3 and then the equations ofAiprimeBiprime BiprimeCiprime and Ai

primeCiprime can be computed by equations (6) (7)

and (8) respectively

y k1x + b1 (6)

where k1 (yi2 minus yi1xi2 minus xi1) and b1 (xi2yi1 minus xi1yi2xi2 minus xi1)

y k2x + b2 (7)

where k2 (yi3 minus yi2xi3 minus xi2) and b2 (xi3yi2 minus xi2yi3xi3minus xi2)

y k3x + b3 (8)


34e Volume Calculation of Voxel After the equations ofthe top surface and the projected triangle edges are calcu-lated the volume of voxel σi can be computed using integralmethod

As shown in Figure 4 the projected triangle is dividedinto two parts by the blue line L It is corresponding that thepentahedron is divided into two parts by a plane that isperpendicular to the xOy plane and passes through the lineL e two parts can be calculated by equations (9) and (10)respectively en we can compute the volume of voxel σiwhich is shown in equation (11)

vi1 1113946xi2

xi1

dx 1113946k1x+b1

k3x+b3

dy 1113946aprimex+bprimey+cprime

0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (9)

vi2 1113946xi3

xi2

dx 1113946k2x+b2

k3x+b3


0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (10)

vi vi1 + vi2 (11)

Let δ k3xi2 + b3 as the red part that is shown inFigure 4 By solving these integral equations we can get thefollowing equations

vi1 aprime yi2 minus δ( 1113857

3

6 k2 minus k3( 11138572 minus

aprimexi2 + cprime( 1113857 yi2 minus δ( 11138572

2 k2 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i21113872 1113873

6k2minus

bprime y3i3 minus δ31113872 1113873

6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(12)

vi2 minusaprime yi2 minus δ( 1113857

3

6 k1 minus k3( 11138572 +


2 k1 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i11113872 1113873

6k1+


6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(13)

In Section 33 we limit the condition of xi1 lt xi2 lt xi3 Ifxi1 xi2 or xi2 xi3 then there is vi1 0 or vi2 0respectively

4 Triangulation-Driven Volume Calculation

is section introduces the DTVC algorithm DTVCfinds triangles one by one without repetition in the 3Dpoint cloud With the discovery of the triangle thevolume of its projected pentahedron is calculated at thesame time In this algorithm we need not to create atriangular mesh and a very large storage structure oftriangle set which can save storage space and time in theactual scene

41 Delaunay Triangulation Delaunay algorithm [41] is apopular used method to construct triangulation meshSuppose the point set is V and T is any triangle of V If T isa Delaunay triangle of V only the interior of the cir-cumcircle of each triangle in T does not contain any pointin V which is called the empty circumcircle principle epoint-by-point insertion algorithm [42 43] also called theBowyerndashWatson algorithm is the most widely usedDelaunay triangulation algorithm

Firstly a large triangle is built to contain all thepoints which is called the supertriangle en insert apoint into the triangular which is connected with thethree vertices of the triangle to form three new trianglesey are checked one by one according to the emptycircumcircle principle Repeat these steps until all pointsare checked

A

B

Aprime

BprimeCprime

C

Figure 2 e triangular mesh and voxel Triangular mesh isgenerated after triangulation of point cloud e triangleAprimeBprimeCprimeon the base plane is the projection of original triangleABC in thetriangular mesh e pentahedron ABC minus AprimeBprimeCprime is called a voxel


42 Supertriangle Creation e supertriangular is a specialtriangle which contains all the points in V is sectionintroduces the creation of the supertriangular

Firstly find the maximum and minimum coordinates ofall points in x and y directions which are denoted asxmax xmin ymax and ymin ese four values construct arectangle which contains all the points in V e rectangle isthe minimum bounding box of all points which is shown asgreen rectangle in Figure 5

Secondly the bounding box is inflated for ϵ length onfour directions to form the red rectangle ABC D Accordingto equation (14) the coordinates of A B C and D are(u t) (v t) (v s) and (u s) respectively Some points maybe on the edge of the green rectangle but all points are insideof the red rectangle

u xmin minus ε

v xmax + ε

s ymin minus ε

t ymax + ε

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

where ε is a very small positive number

irdly extend C D to E which satisfies|DE| (12)|C D| Extend DC to F which satisfies|CF| (12)|C D| en the coordinates of E and F are((3u minus v2) s) and ((3v minus u2) s) respectively

Fourthly connect and extend CA and FB which in-tersect at point G en the coordinate of G is((u + v2) 2(t minus s))

Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(a)

Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(b)

Figure 3 Rearrange the order of Ai Bi and Ci so that xi1 lexi2 le xi3 where Ai Bi and Ci are the vertices of voxel σi and(xi1 yi1 zi1) (xi2 yi2 zi2) (xi3 yi3 zi3) are the coordinates of Ai Bi Ci respectively (a) Bi

prime is on the right of the line AiprimeCiprime in the base plane

(b) Biprime is on the left of the line Ai

primeCiprime in the base plane

L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(a)

L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(b)

Figure 4 e projection in the base plane (a) Biprime is above the line Ai

primeCiprime (b) Bi

prime is under the line AiprimeCiprime

E (3u ndash v2 s) F (3v ndash u2 s)

G (u + v2 2 (t ndash s))

D (u s)

B (v t)A (u t)

C (v s)

εε

Figure 5 e supertriangle e green rectangle is the minimumbounding box of all points e red rectangle is obtained by thebounding box inflating for ϵ length on four directions e bluetriangle is the supertriangle


e blue triangle EFG is the supertriangle

43 Triangulation-Driven Algorithm e key of triangula-tion is to determine whether a triangle is a Delaunay triangleIn Figure 6(a) AB D AC D and BC D are unde-termined triangles When inserting a point E we makecircumcircle for each undetermined triangle as shown inFigure 6(b) According to the relative position of E andcircumcircle we can determine the triangle type e de-cision rules are as follows

(i) Point E is on the right of circumcircle of BC D

(green circle) and then BC D is a Delaunaytriangle

(ii) Point E is in the circumcircle of AC D (bluecircle) and thenAC D is not a Delaunay triangleDelete AC D from the undetermined trianglesand A DE ACE and C DE are new unde-termined triangles

(iii) Point E is outside and not on the right of the cir-cumcircle ofAB D (red circle) and thenAB D

is still an undetermined triangle

In the DTVC algorithm instead of storing the Delaunaytriangles we directly calculate the projected volume ealgorithm is shown in Algorithm 1is algorithm is used tofind triangle without repetition We do not create and storethe mesh in the calculation When all the points in the 3Dpoint cloud are traversed the volume is obtained

ere is a shared algorithm to calculate volume of a setwhich is shown in Algorithm 1 e calculation of this al-gorithm is based on equations (12) and (13)

5 Experiments

51 Datasets In order to analyze the error of the algorithmwe simulate two point cloud data as shown in Figure 7Figure 7(a) has a single peak whereas Figure 7(b) hasmultiple peaks e single and multiple peaks have 10201and 40401 points respectively

To test the volume calculation in real scene we take 59photos in a 320 times 110 times 49 coal shed to generate the pointcloud e resolution of the photos is 3072 times 2048

52 Experiment Setup In error analysis we use MatlabR2018b implementation toole processor andmemory areIntel Core i7-8565U and 160GB respectivelye operatingsystem is Win10

In the real scene the 3D reconstruction and volumecalculation system run on Ubuntu 1804 LTS e CPUGPU and memory are Xeon Gold 6226R Nvidia 16GBTesla T4 and 256G DDR4 respectively

e error is calculated by equation (15) and the CUPtime is the difference between the finish time and the starttime of the algorithm We repeat each algorithm 100 timesand average the errors and CUP times as the final experi-ment result

Err v minus vr

11138681113868111386811138681113868111386811138681113868

vr

(15)

where v and vr are the measured and the real volumerespectively

53ComparewithState-of-the-ArtMethod We choose somepopular used methods to compare with our proposedmethod e notation for these methods is as follows

(i) Slice- [29] Slicing method and ldquordquo is the slicenumber

(ii) P-T3 [17] e projection method which divides thepentahedron into 3 tetrahedrons

(iii) DTVC Our Delaunay triangulation-driven volumecalculation

Tables 1 and 2 show the result of single peak andmultiplepeaks respectively Experimental results show that theslicing method has the most efficient operation Howeverthe error rate is also large When the slice number is in-creased the error goes down and the running time rises upe results based on the slicing model are difficult to beapplied to practical solutions

P-T3 and our DTVC have the same low error but P-T3 ismore time-consuming than DTVC because P-T3 requiresthree determinant calculations of matrix P-T3 spends morethan 2 times on the running time evaluation

Experimental results show that our DTVC method hasachieved a good balance between error and efficiency

54 Application in Real Scene In a real scene we deploy acamera matrix in a coal shed We generate the point cloudusing the 3D movement structure recovery method Basedon the point cloud we use the proposed DTVC method tocalculate the volume of coal pile Figure 8 shows a displayinterface of the system

6 Conclusion

In the volume measurement of nonconvex point cloud thetriangulation projection method has advantages in accuracyand efficiency Traditional method of decomposing voxelinto tetrahedrons is concise However experimental resultsshow that the method of finding analytic solution for voxelvolume integral is complex but more efficient erefore theanalytic solution method is better than the divide-and-conquer method in efficiency


1

08

06

04

02

01

050

ndash05ndash1

105

0ndash05

ndash1

(a)

50

40

30

20

10

02

10

ndash1ndash2 210ndash1ndash2

(b)

Figure 7 Test data for error analysis e real volumes of (a) and (b) are 29284 and 3626667 respectively (a) Single peak (b) Multiplepeaks

A

B

C

DE

(a)

A

B

C

D

E

(b)

Figure 6 Delaunay triangulation When point E is inserted the undetermined triangles are checked by circumcircle testing (a) Un-determined triangles (b) Circumcircle test

(i) Input the vertex list V

(ii) Output the volume Vol of V

(1) Vol 0(2) Initialize the temporary triangle list T(3) Calculate the supertriangle EFG from V(4) V middot Add(E) V middot Add(F) V middot Add(G)(5) T middot Add(EFG)(6) foreachPinVdo(7) Initialize the edge buffer Edge(8) foreachABCinTdo(9) Calculate the triangle circumcircle center O and radius r(10) ifP middot xgtO middot x + rthen(11) Vol+ Algorithm 2(T E F G )

ALGORITHM 1 Continued


(12) end(13) if|OP|le rthen(14) Edge middot Add(AB)Edge middot Add(BC)Edge middot Add(AC)(15) T middot Delete(ABC)(16) end(17) end(18) Delete all doubly specified edges from Edge(19) foreachABinEdgedo(20) T middot Add(ABP)(21) end(22) end(23) Vol+ Algorithm 2(T E F G )

ALGORITHM 1 Delaunay triangulation-driven volume calculation

(i) Input the triangle list T and the excluded point set P

(ii) Output the summary volume Vol of T where the triangle is excluded in which one of vertices is in P

(1) Vol 0(2) foreachABCinTdo(3) ifA notin P andB notin P andC notin Pthen(4) Calculate the projection volume vi of ABC(5) Vol+ vi(6) end(7) end

ALGORITHM 2 Triangle projection volume calculation

Table 1 e single peak result

Method Volume Error rate () Running time (ms)Slice-20 32290 1026 12937Slice-50 31681 819 13309Slice-100 31495 755 15125Slice-1000 30711 487 55491P-T3 29282 00068 162699Our DTVC 29282 00068 75298e real volume is 29284

Table 2 e multiple peaks result



Data Availability

e data used to support the findings of this study areavailable at httpshttpwwwdropboxcomshmce9q1n9znm8y22AAB7cCqEPvD4Vl2Z_JxROP_ladl0

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Project of Coal Measure-ment of Huadian Zibo ermal Power Co Ltd

References

[1] W Zhang D Yang Y Li and W Xu Portable 3D LaserScanner for Volume Measurement of Coal Pile pp 340ndash347Springer Singapore Singapore 2020

[2] Y Huang K Zheng X Liu H Wang and J Ruan Appli-cation of 3d Laser Scanner in Ore Volume MeasurementScience of Surveying and Mapping Beijing China 2012

[3] F Jia J Li C Wang Y Yu and D Zai ldquoEarthwork volumesestimation in asphalt pavement reconstruction using a mobilelaser scanning systermrdquo in Proceedings of the Geoscience ampRemote Sensing Symposium Quebec City Canada July 2014

[4] X Guo F Wang Y Ma and D Du ldquoResearch on three-dimensional point clouds processing for standing tree volumebased on laser scannerrdquo in Proceedings of the 2009 SecondInternational Symposium on Knowledge Acquisition andModeling pp 206ndash208 Wuhan China December 2009

[5] X Wei Y Wang J Zheng M Wang and Z Feng ldquoTreecrown volume calculation based on 3-d laser scanning pointclouds datardquo Transactions of the Chinese Society for Agri-cultural Machinery vol 44 pp 235ndash240 2013

[6] A Basher B C Kim K H Lee and H Y Jung ldquoAutomaticlocalization and discrete volume measurements of hippo-campi from MRI data using a convolutional neural networkrdquoIEEE Access vol 8 pp 91725ndash91739 2020

[7] D Hadhazi B Czetenyi A Horvath G Orban G Horvathand A Horvath ldquoLung nodule volume measurement usingdigital chest tomosynthesisrdquo in Proceedings of the 2015 IEEEInternational Instrumentation and Measurement Technology

Conference (I2MTC) Proceedings pp 2026ndash2031 Pisa ItalyMay 2015

[8] C Okinda Y Sun I Nyalala et al ldquoEgg volume estimationbased on image processing and computer visionrdquo Journal ofFood Engineering vol 283 p 10 2020

[9] Z Cheng M Yang B Kang X Bian and J Cui ldquoFeatureparameter extraction algorithm for the large-scale complexstructure tank based on 3d laser scanning volume measure-mentrdquo in Proceedings of the 2019 14th IEEE InternationalConference on Electronic Measurement amp Instruments(ICEMI) pp 651ndash657 Changsha China November 2019

[10] H Hao X Liang X Chen H Shi and P Yi ldquoe automaticmeasurement system of large vertical storage tank volumebased on 3d laser scanning principlerdquo in Proceedings of the2017 13th IEEE International Conference on Electronic Mea-surement amp Instruments (ICEMI) Yangzhou China October2017

[11] T Suzuki K Futatsuishi and K Kobayashi ldquoFood volumeestimation using 3d shape approximation for medicationmanagement supportrdquo in Proceedings of the 2018 3rd Asia-Pacific Conference on Intelligent Robot Systems (ACIRS)pp 107ndash111 Singapore Singapore July 2018

[12] R L Graham ldquoAn efficient algorithm for determining theconvex hull of a finite planar setrdquo Information ProcessingLetters vol 1 no 4 pp 73ndash82 1972

[13] Z Xu and H Xu ldquoFast algorithm of computing volume basedon convex hullrdquo Computer Engineering and Applicationsvol 29 no 4 2013

[14] Y Lu and J Wang ldquoStudy on reslicing method for volumedata from industrial ctrdquo Computer Engineering amp Applica-tions vol 43 no 22 pp 201ndash203 2007

[15] D Sen and T K Srikanth ldquoEfficient computation of volumefractions for multi-material cell complexes in a grid by slic-ingrdquo Computers amp Geosciences vol 34 no 7 pp 754ndash7822008

[16] X Lv Y Miao X Ren J Wu M Zhang and Y Gu ldquoestudy and implementation of liver volume measuring methodbased on 3-dimensional reconstruction technologyrdquo Optikvol 126 no 17 pp 1534ndash1539 2015

[17] Y Hu J Wang Y Fan Y Lu and J Zhang ldquoLidar-basedthree-dimensional modeling and volume calculation for spaceobjectsrdquo Chinese Journal of Lasers vol 47 no 5 Article ID0510001 2020

[18] W Wang H Duan D Zhang and R Jia ldquoA method forsolving the volume and surface area of irregular body afterthree-dimensional laser scanningrdquo Geomatics amp Spatial In-formation Technology vol 42 pp 172ndash175 2019

[19] M Wu X Yi H Luo C Li X Tang and K Chen ldquoOn-linemeasurement method for volume and surface area of redjujube based on multi-contour modelrdquo Transactions of theChinese Society of Agricultural Engineering vol 35 pp 283ndash290 2019

[20] E Speakman and G Averkov ldquoComputing the volume of theconvex hull of the graph of a trilinear monomial using mixedvolumesrdquo Discrete Applied Mathematics In press 2019

[21] YWu A Gupta K Kurzeja and J Rossignac ldquoChocc convexhull of cospherical circles and applications to latticesrdquoComputer-Aided Design vol 129 Article ID 102903 2020

[22] R Alshamrani F Alshehri and H Kurdi ldquoA preprocessingtechnique for fast convex hull computationrdquo ProcediaComputer Science vol 170 pp 317ndash324 2020

[23] J Zhao L Jiao F Liu et al ldquo3d fast convex-hull-basedevolutionary multiobjective optimization algorithmrdquo AppliedSoft Computing vol 67 pp 322ndash336 2018

Figure 8 Volume calculation in a real scene


[24] F A Auat Cheein and J Guivant ldquoSlam-based incrementalconvex hull processing approach for treetop volume esti-mationrdquo Computers and Electronics in Agriculture vol 102pp 19ndash30 2014

[25] Y-J Kim M-S Kim and G Elber ldquoPrecise convex hullcomputation for freeform models using a hierarchical gaussmap and a coons bounding volume hierarchyrdquo Computer-Aided Design vol 46 pp 252ndash257 2014

[26] K Song and S Ou ldquoYung-ping-tien a client-server archi-tecture for object volumemeasurement on a conveyor beltrdquo inProceedings of the 201912th Asian Control Conference (ASCC)pp 901ndash906 Kitakyushu Japan June 2019

[27] N Bandi R B Tunyogi Z Szabo E Farkas and C SulyokldquoImage-based volume estimation using stereo visionrdquo inProceedings of the 2020 IEEE 18th International Symposium onIntelligent Systems and Informatics (SISY) pp 55ndash60 Subo-tica Serbia September 2020

[28] S M Shah J P Crawshaw F Gray J Yang and E S BoekldquoConvex hull approach for determining rock representativeelementary volume for multiple petrophysical parametersusing pore-scale imaging and Lattice-Boltzmann modellingrdquoAdvances in Water Resources vol 104 pp 65ndash75 2017

[29] W Chang C Wu Y Tsai and W Chiu ldquoObject volumeestimation based on 3d point cloudrdquo in Proceedings of the2017 International Automatic Control Conference (CACS)pp 1ndash5 Pingtung Taiwan November 2017

[30] J Li G Liu and Y Liu ldquoA dynamic volume measurementsystem with structured light visionrdquo in Proceedings of the 201631st Youth Academic Annual Conference of Chinese Associ-ation of Automation (YAC) pp 251ndash255 Wuhan ChinaNovember 2016

[31] F Heckel H Meine J H Moltz et al ldquoSegmentation-basedpartial volume correction for volume estimation of solid le-sions in ctrdquo IEEE Transactions on Medical Imaging vol 33no 2 pp 462ndash480 2014

[32] J Avdal A Rodriguez-Molares E Andreas Rye Berg andH Torp ldquoVolume flow estimation in valvular jets using 3dhigh frame rate ultrasoundrdquo in Proceedings of the 2018 IEEEInternational Ultrasonics Symposium (IUS) pp 1ndash4 KobeJapan October 2018

[33] M Lyra J Striligas M Gavrilelli C Chatzijiannis andK Skouroliakou ldquoyroid volume determination by singlephoton tomography and 3d processing for activity dose es-timationrdquo in Proceedings of the 2008 IEEE InternationalWorkshop on Imaging Systems and Techniques pp 17ndash20Chania Greece September 2008

[34] M Yu J Liou S Kuo M Lee and Y Hung ldquoNoncontactrespiratory measurement of volume change using depthcamerardquo in Proceedings of the 2012 Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 2371ndash2374 San Diego CA USA September 2012

[35] E B Putman and S C Popescu ldquoAutomated estimation ofstanding dead tree volume using voxelized terrestrial lidardatardquo IEEE Transactions on Geoscience and Remote Sensingvol 56 no 11 pp 6484ndash6503 2018

[36] Y Gao H Zhang Y Zhu et al ldquoVoxel-based partial volumecorrection of amyloid pet images incorporating non-localmeans regularizationrdquo in Proceedings of the 2018 IEEE Nu-clear Science Symposium and Medical Imaging ConferenceProceedings (NSSMIC) pp 1ndash4 Sydney Australia November2018

[37] M Aronsson ldquoEstimating fibre twist and aspect ratios in 3dvoxel volumesrdquo in Proceedings of the Object recognition

supported by user interaction for service robots pp 218ndash221Quebec City Canada August 2002

[38] C Potena R F Carpio N Pietroni et al ldquoSuckers emissiondetection and volume estimation for the precision farming ofhazelnut orchardsrdquo in Proceedings of the 2020 IEEE Confer-ence on Control Technology and Applications (CCTA)pp 285ndash290 Montreal Canada August 2020

[39] P Toumpaniaris I Skalkidis A Giakoumaki andD Koutsouris ldquoVolume estimation of non-geometric shapecavity using an array of normal distributed distance sensorson a spherical mount applicable in the right ventriclerdquo inProceedings of the 2009 9th International Conference on In-formation Technology and Applications in Biomedicinepp 1ndash4 Larnaka Cyprus November 2009

[40] Z Cai C Jin J Xu and T Yang ldquoMeasurement of potatovolume with laser triangulation and three-dimensional re-constructionrdquo IEEE Access vol 8 pp 176565ndash176574 2020

[41] C L Lawson ldquoSoftware for c1 surface interpolationrdquo inMathematical Software pp 161ndash194 University ofWisconsinMadison WI USA 1977

[42] A Bowyer ldquoComputing dirichlet tessellationsrdquoeComputerJournal vol 24 no 2 pp 162ndash166 1981

[43] D F Watson ldquoComputing the n-dimensional delaunay tes-sellation with application to voronoi polytopesrdquo e Com-puter Journal vol 24 no 2 pp 167ndash172 1981


the diagonal lines of each pentahedron face and then thepentahedron is divided into three tetrahedrons e volumeof pentahedron is obtained by accumulating the volumes oftetrahedrons [17] (2) e pentahedron is approximated to atriangular prism to calculate the volume [18]ese methodshave large error or are time-consuming

is paper proposes a Delaunay triangulation-drivenvolume calculation (DTVC)method where the two stages inthe triangulation projection method are merged togetherWith the construction of triangular mesh the volume cal-culation is completed e pentahedron volume calculationis also improved It is more accurate and efficient than othermethods

We summarize our contributions as follows

(i) An accurate and direct method is proposed tocalculate the volume of the pentahedron whichprovides the analytic solutions of pentahedron us-ing the integral method

(ii) We also combined the two stages to one for thetriangulation projection method e volume cal-culation is driven by the triangulation

(iii) e calculation error of the whole point cloudvolume is less than the slicing method and theefficiency is higher than the existing projectionmethod e proposed method has achieved a goodbalance between error and efficiency

e remainder of this paper is organized as followsSection 2 discusses the related work about the volumecalculation of point cloud In Section 3 the accurate anddirect pentahedron volume calculation is presented InSection 4 the triangulation-driven volume calculation isintroduced Section 5 provides the experimental resultsSection 6 concludes this paper

2 Related Work

is section discusses the volume calculation methods

21 RoughMethods Guo et al [4] calculate the standing treevolume using an empirical formula where the volume is theproduct of the cross-sectional area at breast height theheight plus three and a regulatory factor Wu et al [19] andOkinda et al [8] regard the jujubes and eggs as ellipsoids tocalculate the volume In [11] eachmass of food on the dish islooked as a cylinder In [9 10] the tank is approximated to a

cylinder Hadhazi et al [7] place 16 simulated small ellipticalnodules into an anthropomorphic chest phantom to cal-culate the lung nodule volume Because the real objects arenot standard geometry the error is very large for thesemethods

22 Convex Hull Method In 1972 Graham proposed theconvex hull method [12] Speakman and Averkov [20] in-troduce the convex hull volume formula of a trilinearmonomial Wu et al [21] discuss the properties and com-putation of the boundary of convex hulls of cosphericalcircles Alshamrani et al [22] filter all points inside a four-vertex polygon to reduce the computational cost for com-puting a convex hull for a large set of points Zhao et al [23]introduce an incremental convex hull calculation and a fastreplacement for nondominated sorting Auat Cheein andGuivant [24] split the 3D raw data into subsets which havethe same convex hull Calculating the convex hull fromsubsets can decrease the computational cost of the entireprocess Kim et al [25] propose a precise convex hullcomputation method for free form models where a Gaussmap organized in a hierarchy of normal pyramids and aCoons bounding volume hierarchy are prebuilt

In application to calculate the object volume on aconveyor belt Song and Ou [26] apply a convex hull al-gorithm to find the set of the smallest convex polygon thatcontains all the object point cloud Furthermore they find arectangle which has the smallest area to determine the objectsize Bandi et al [27] also use the convex hull method tocalculate the volume of dense point cloud Xu and Xu [13]use the incremental algorithm to calculate the convex hull ofthe point cloud to approximate the object Shah et al [28]apply convex hull to calculate the representative element ofvolume by simultaneously considering the two main mac-roscopic petrophysical parameters porosity and single-phase permeability

23 SlicingMethod Chang et al [29] cut the point cloud intoslices of equal thickness along the z-axis en divide eachslice along the y-axis equally and cut each slice into sub-intervals along the x-axis Using the minimum and maxi-mum coordinates in each subinterval to calculate the area ofeach sliceen the slice areas are integrated along the z-axisto estimate the volume of the object Li et al [30] take aframe in the structured light vision as a slice Heckel et al[31] propose a segmentation-based partial volume analysis

Figure 1 Coal pile from different view positions It is not a convex model












V 1113944n

i1vi (1)





nirarr

irarr

jrarr

krarr



111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(2)

nirarr

(a b c) (3)


















y k1x + b1 (6)


y k2x + b2 (7)


y k3x + b3 (8)




vi1 1113946xi2

xi1

dx 1113946k1x+b1

k3x+b3


0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (9)

vi2 1113946xi3

xi2

dx 1113946k2x+b2

k3x+b3


0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (10)

vi vi1 + vi2 (11)



3



2 k2 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i21113872 1113873

6k2minus


6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(12)


3

6 k1 minus k3( 11138572 +


2 k1 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i11113872 1113873

6k1+


6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(13)






A

B

Aprime

BprimeCprime

C






u xmin minus ε

v xmax + ε

s ymin minus ε

t ymax + ε

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)




Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(a)

Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(b)





L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(a)

L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(b)


primeCiprime (b) Bi




D (u s)

B (v t)A (u t)

C (v s)

εε












5 Experiments






Err v minus vr

11138681113868111386811138681113868111386811138681113868

vr

(15)










6 Conclusion



1

08

06

04

02

01

050

ndash05ndash1

105

0ndash05

ndash1

(a)

50

40

30

20

10

02

10


(b)


A

B

C

DE

(a)

A

B

C

D

E

(b)


















Data Availability




Acknowledgments


References



























































V 1113944n

i1vi (1)





nirarr

irarr

jrarr

krarr



111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(2)

nirarr

(a b c) (3)


















y k1x + b1 (6)


y k2x + b2 (7)


y k3x + b3 (8)




vi1 1113946xi2

xi1

dx 1113946k1x+b1

k3x+b3


0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (9)

vi2 1113946xi3

xi2

dx 1113946k2x+b2

k3x+b3


0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (10)

vi vi1 + vi2 (11)



3



2 k2 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i21113872 1113873

6k2minus


6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(12)


3

6 k1 minus k3( 11138572 +


2 k1 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i11113872 1113873

6k1+


6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(13)






A

B

Aprime

BprimeCprime

C






u xmin minus ε

v xmax + ε

s ymin minus ε

t ymax + ε

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)




Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(a)

Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(b)





L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(a)

L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(b)


primeCiprime (b) Bi




D (u s)

B (v t)A (u t)

C (v s)

εε












5 Experiments






Err v minus vr

11138681113868111386811138681113868111386811138681113868

vr

(15)










6 Conclusion



1

08

06

04

02

01

050

ndash05ndash1

105

0ndash05

ndash1

(a)

50

40

30

20

10

02

10


(b)


A

B

C

DE

(a)

A

B

C

D

E

(b)


















Data Availability




Acknowledgments


References


























































y k1x + b1 (6)


y k2x + b2 (7)


y k3x + b3 (8)




vi1 1113946xi2

xi1

dx 1113946k1x+b1

k3x+b3


0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (9)

vi2 1113946xi3

xi2

dx 1113946k2x+b2

k3x+b3


0dz

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 (10)

vi vi1 + vi2 (11)



3



2 k2 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i21113872 1113873

6k2minus


6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(12)


3

6 k1 minus k3( 11138572 +


2 k1 minus k3( 1113857

111386811138681113868111386811138681113868111386811138681113868

+bprime y

3i3 minus y

3i11113872 1113873

6k1+


6k3

1113868111386811138681113868111386811138681113868111386811138681113868

(13)






A

B

Aprime

BprimeCprime

C






u xmin minus ε

v xmax + ε

s ymin minus ε

t ymax + ε

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)




Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(a)

Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(b)





L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(a)

L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(b)


primeCiprime (b) Bi




D (u s)

B (v t)A (u t)

C (v s)

εε












5 Experiments






Err v minus vr

11138681113868111386811138681113868111386811138681113868

vr

(15)










6 Conclusion



1

08

06

04

02

01

050

ndash05ndash1

105

0ndash05

ndash1

(a)

50

40

30

20

10

02

10


(b)


A

B

C

DE

(a)

A

B

C

D

E

(b)


















Data Availability




Acknowledgments


References




















































u xmin minus ε

v xmax + ε

s ymin minus ε

t ymax + ε

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)




Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(a)

Ai (xi1 yi1 zi1)

Bi (xi2 yi2 zi2)

Bprimei (xi2 yi2 0)

Ci (xi3 yi3 zi3)

Cprimei (xi3 yi3 0)

Aprimei (xi1 yi1 0)

(b)





L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(a)

L

Aprimei (xi1 yi1)

Bprimei (xi2 yi2)

Cprimei (xi3 yi3)

y

x0

δ

(b)


primeCiprime (b) Bi




D (u s)

B (v t)A (u t)

C (v s)

εε












5 Experiments






Err v minus vr

11138681113868111386811138681113868111386811138681113868

vr

(15)










6 Conclusion



1

08

06

04

02

01

050

ndash05ndash1

105

0ndash05

ndash1

(a)

50

40

30

20

10

02

10


(b)


A

B

C

DE

(a)

A

B

C

D

E

(b)


















Data Availability




Acknowledgments


References


























































5 Experiments






Err v minus vr

11138681113868111386811138681113868111386811138681113868

vr

(15)










6 Conclusion



1

08

06

04

02

01

050

ndash05ndash1

105

0ndash05

ndash1

(a)

50

40

30

20

10

02

10


(b)


A

B

C

DE

(a)

A

B

C

D

E

(b)


















Data Availability




Acknowledgments


References

















































1

08

06

04

02

01

050

ndash05ndash1

105

0ndash05

ndash1

(a)

50

40

30

20

10

02

10


(b)


A

B

C

DE

(a)

A

B

C

D

E

(b)


















Data Availability




Acknowledgments


References




























































Data Availability




Acknowledgments


References

















































accurate volume calculation driven by delaunay

Documents